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<li><a class="reference internal" href="#">Curve Fitting with Bayesian Ridge Regression</a></li>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-linear-model-plot-bayesian-ridge-curvefit-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="curve-fitting-with-bayesian-ridge-regression">
<span id="sphx-glr-auto-examples-linear-model-plot-bayesian-ridge-curvefit-py"></span><h1>Curve Fitting with Bayesian Ridge Regression<a class="headerlink" href="#curve-fitting-with-bayesian-ridge-regression" title="Permalink to this headline">¶</a></h1>
<p>Computes a Bayesian Ridge Regression of Sinusoids.</p>
<p>See <a class="reference internal" href="../../modules/linear_model.html#bayesian-ridge-regression"><span class="std std-ref">Bayesian Ridge Regression</span></a> for more information on the regressor.</p>
<p>In general, when fitting a curve with a polynomial by Bayesian ridge
regression, the selection of initial values of
the regularization parameters (alpha, lambda) may be important.
This is because the regularization parameters are determined by an iterative
procedure that depends on initial values.</p>
<p>In this example, the sinusoid is approximated by a polynomial using different
pairs of initial values.</p>
<p>When starting from the default values (alpha_init = 1.90, lambda_init = 1.),
the bias of the resulting curve is large, and the variance is small.
So, lambda_init should be relatively small (1.e-3) so as to reduce the bias.</p>
<p>Also, by evaluating log marginal likelihood (L) of
these models, we can determine which one is better.
It can be concluded that the model with larger L is more likely.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Author: Yoshihiro Uchida &lt;nimbus1after2a1sun7shower@gmail.com&gt;</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>

<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">BayesianRidge</span>


<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>


<span class="c1"># #############################################################################</span>
<span class="c1"># Generate sinusoidal data with noise</span>
<span class="n">size</span> <span class="o">=</span> <span class="mi">25</span>
<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">1234</span><span class="p">)</span>
<span class="n">x_train</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">size</span><span class="p">)</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x_train</span><span class="p">)</span> <span class="o">+</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
<span class="n">x_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>


<span class="c1"># #############################################################################</span>
<span class="c1"># Fit by cubic polynomial</span>
<span class="n">n_order</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vander</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">n_order</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">increasing</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vander</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">n_order</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">increasing</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># Plot the true and predicted curves with log marginal likelihood (L)</span>
<span class="n">reg</span> <span class="o">=</span> <span class="n">BayesianRidge</span><span class="p">(</span><span class="n">tol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">compute_score</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ax</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">axes</span><span class="p">):</span>
    <span class="c1"># Bayesian ridge regression with different initial value pairs</span>
    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
        <span class="n">init</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_train</span><span class="p">),</span> <span class="mf">1.</span><span class="p">]</span>  <span class="c1"># Default values</span>
    <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
        <span class="n">init</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1e-3</span><span class="p">]</span>
        <span class="n">reg</span><span class="o">.</span><span class="n">set_params</span><span class="p">(</span><span class="n">alpha_init</span><span class="o">=</span><span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">lambda_init</span><span class="o">=</span><span class="n">init</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
    <span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
    <span class="n">ymean</span><span class="p">,</span> <span class="n">ystd</span> <span class="o">=</span> <span class="n">reg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">return_std</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>

    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">func</span><span class="p">(</span><span class="n">x_test</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;blue&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;sin($2</span><span class="se">\\</span><span class="s2">pi x$)&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;observation&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">ymean</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;red&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;predict mean&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">x_test</span><span class="p">,</span> <span class="n">ymean</span><span class="o">-</span><span class="n">ystd</span><span class="p">,</span> <span class="n">ymean</span><span class="o">+</span><span class="n">ystd</span><span class="p">,</span>
                    <span class="n">color</span><span class="o">=</span><span class="s2">&quot;pink&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;predict std&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">1.3</span><span class="p">,</span> <span class="mf">1.3</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
    <span class="n">title</span> <span class="o">=</span> <span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">alpha$_init$=</span><span class="si">{:.2f}</span><span class="s2">,</span><span class="se">\\</span><span class="s2"> </span><span class="se">\\</span><span class="s2">lambda$_init$=</span><span class="si">{}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
            <span class="n">init</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">init</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
        <span class="n">title</span> <span class="o">+=</span> <span class="s2">&quot; (Default)&quot;</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
    <span class="n">text</span> <span class="o">=</span> <span class="s2">&quot;$</span><span class="se">\\</span><span class="s2">alpha=</span><span class="si">{:.1f}</span><span class="s2">$</span><span class="se">\n</span><span class="s2">$</span><span class="se">\\</span><span class="s2">lambda=</span><span class="si">{:.3f}</span><span class="s2">$</span><span class="se">\n</span><span class="s2">$L=</span><span class="si">{:.1f}</span><span class="s2">$&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
           <span class="n">reg</span><span class="o">.</span><span class="n">alpha_</span><span class="p">,</span> <span class="n">reg</span><span class="o">.</span><span class="n">lambda_</span><span class="p">,</span> <span class="n">reg</span><span class="o">.</span><span class="n">scores_</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.05</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">text</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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